let r be Real; :: thesis: for seq being Real_Sequence st 0 <= r & ( for n being Nat holds seq . n = 1 / ((n * n) + r) ) holds
seq is convergent
let seq be Real_Sequence; :: thesis: ( 0 <= r & ( for n being Nat holds seq . n = 1 / ((n * n) + r) ) implies seq is convergent )
assume that
A1: 0 <= r and
A2: for n being Nat holds seq . n = 1 / ((n * n) + r) ; :: thesis: seq is convergent
take 0 ; :: according to SEQ_2:def 6 :: thesis: for b₁ being object holds
( b₁ <= 0 or ex b₂ being set st
for b₃ being set holds
( not b₂ <= b₃ or not b₁ <= |.((seq . b₃) - 0).| ) )
let p be Real; :: thesis: ( p <= 0 or ex b₁ being set st
for b₂ being set holds
( not b₁ <= b₂ or not p <= |.((seq . b₂) - 0).| ) )
assume 0 < p ; :: thesis: ex b₁ being set st
for b₂ being set holds
( not b₁ <= b₂ or not p <= |.((seq . b₂) - 0).| )
then consider n being Nat such that
A3: for m being Nat st n <= m holds
|.((seq . m) - 0).| < p by A1, A2, LmTh32;
take n ; :: thesis: for b₁ being set holds
( not n <= b₁ or not p <= |.((seq . b₁) - 0).| )
thus for b₁ being set holds
( not n <= b₁ or not p <= |.((seq . b₁) - 0).| ) by A3; :: thesis: verum